The dynamics of \(\Psi \)  linearizes if and only if there exists a scalar \(\lambda \)  that relates the control cost to the covariance matrix such that \(B=\lambda K^{-1}\) . This criteria is known as the path-integral control condition [1]}, [2]} by which we can map a generally non-linear control problem onto a linear stochastic process. The path-integral control condition implies that the cost of artificial selection on each phenotype should be inversely proportional to the phenotype's fluctuations. In other words, artificially tweaking with highly conserved phenotypes should be more costly than with variable phenotypes. In this case, the HJB equation for the transformed cost-to-go function \(\Psi \)  follows,
\(\frac{\partial }{\partial t} \Psi = - A({\bf x})^\top \cdot \nabla \Psi - \frac{1}{2} {\text{\bf Tr}}K \nabla _{{\bf x}{\bf x}} \Psi + \frac{1}{\lambda } V({\bf x}) \Psi \equiv - { L}^\dagger \Psi \) 
